Studying nanotube-based oscillators and their application as memory cells via nanoscale continuum modeling and simulation

Shaoping Xiao et al. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 5, Issue 12, (Part - 2) December 2015, pp.160-170
www.ijera.com 160 | P a g e
Studying nanotube-based oscillators and their application as
memory cells via nanoscale continuum modeling and simulation
Shaoping Xiao*, Weixuan Yang**, Yan Zhang***
*(Department of Mechanical and Industrial Engineering, College of Engineering, The University of Iowa, USA)
** (Product Development & Global Technology Division, Caterpillar Inc., USA)
*** (SMIT Center, School of Mechatronics Engineering and Automation, Shanghai University, China)
ABSTRACT
A nanoscale continuum model of carbon nanotube-based oscillators is proposed in this paper. In the continuum
model, the nanotube is discretized via the meshfree particle method. The atomistic interlayer interaction between
the outer and inner tubes is approximated by the interlayer interaction between particles. The mechanical
behaviors of oscillators are studied and compared well with molecular dynamics simulation results. The
nanotube-based oscillator can be employed to design a nanoelectromechanical system. In this system, two
electrodes are attached on the top of the outer tube so that the induced electromagnetic force can overcome the
interlayer friction. The mechanisms of such nanoelectromechanical systems as memory cells are also considered.
Keywords - carbon nanotube, oscillator, memory cell, meshfree, nanoscale continuum modeling
I. INTRODUCTION
Recent developments in nanotechnology
promote that nanoscale materials and devices can
enhance new engineering techniques in the relevant
technology areas, including nanoelectromechanical
systems (NEMS). Current advancements in nanotube
technology have occurred predominantly in the area
of carbon nanotube (CNT) development [1, 2] for
nanoscale electronic components and devices [3-7].
Numerical methods play an important role in
engineering design to accelerate and foster
maturation of nanotechnology that will enable future
electronics to become smaller, lighter, multi-
functional and autonomous. Since molecular
dynamics (MD) has been a powerful tool to elucidate
complex physical phenomena at the nanoscale, it is
one of the effective candidates to assist
nanoengineering designs.
Srivastava [8] proposed a phenomenological
design of a single laser-powered molecular motor for
CNT-based gears. Using MD simulations, he found
that the rotational angular momentum of the driven
gear tended to stabilize the rotational dynamics of the
system. Kang and Hwang [9] proposed another
conceptual design of CNT-based motor. They
investigated nanoscale engine schematics composed
of a CNT oscillator, motor, channel, and nozzle via
MD simulations. Popov et al. [10] reported on three
new NEMS designs based on multi-walled carbon
nanotubes (MWNT): an electromechanical
nanothermometer, a nanorelay, and a nanomotor.
They used Ab initio and semi-empirical calculations
to estimate the operational characteristics and
dimensions of their proposed NEMS systems. Zhao
and his co-workers [11] studied CNT devices based
on the crossbar architecture and found that this
topology was well suited for parallel programming or
learning in the context neuromorphic computing.
A few electromechanical models were employed
to study CNT-based devices due to the limitation of
MD in length and time scales. The electromechanical
models can be viewed as continuum models since the
electric fields and the configuration of CNTs were
modeled via continuum approaches. Axelsson et al.
[12] presented theoretical and experimental
investigations of three-terminal NEMS relays based
on suspended CNTs, which were modeled via the
classical continuum elasticity theory. Ke and
Espinosa [13] examined a switchable CNT-based
NEMS with closed-loop feedback. They investigated
the pull-in/pull-out and tunneling characteristics of
the system by means of an electromechanical
analysis, in which the model included the
concentration of electrical charge via an empirical
function, and the van der Waals force via a
continuum approximation. They also used the
assumption of continuum mechanics to derive a
complete nonlinear equation of the elastic line of the
CNT for modeling and simulating CNT-based NEMS
devices [14]. In other studies, Bichoutskaia [15]
employed an electromechanical model to study
interwall interactions in CNTs. Poot et al. [16]
developed a model for flexural oscillations of
suspended CNTs based on the theory of continuum
mechanics.
Recently, a few efficient multiscale methods that
are capable of addressing large length and time scales
have been developed. Multiscale methods either
employ a nanoscale continuum approach to
approximate a group of atoms [17-19] or couple MD
RESEARCH ARTICLE OPEN ACCESS

with continuum mechanics [20-22]. Some researchers
have combined the classical MD simulations with the
continuum electric models to investigate CNT-based
NEMS devices. For example, Kang et al. [23]
employed this combined method to study suspended
CNTs for memory device applications. Dequesnes et
al. [24] reported a combined MD/continuum
simulation of CNT-based NEMS switches. In their
research, a multiscale method that combined the
nonlinear beam theory with MD was presented to
model and simulate CNTs. Xiao and Hou [25]
proposed the first numerical modeling and study of
CNT-based resonant oscillators. In their multiscale
model, the paddle was modeled as the continua while
the CNT was modeled via MD. In addition, the
molecular and continuum domains were attached
with each other at the interfaces via the edge-to-edge
coupling [26].
CNT-based oscillators have been of great interest
to scientists and engineers since Zheng et al. [27, 28]
employed a MWNT to design gigahertz oscillators. A
few MD simulations have been conducted to study
mechanisms of CNT-based oscillators. The
oscillatory frequency of the oscillators might be up to
87 GHz based on MD simulations [29-31]. Legoas et
al. [29] also pointed out that stable oscillators were
only possible when the interlayer distances between
the outer and inner tubes were of ~0.34 nm, the
interlayer distance of regular MWNTs. Unlike other
CNT-based machines, the energy dissipation plays a
key role and needs to be considered when designing a
stable nano-oscillator. Xiao et al. [31] observed
mechanical energy dissipation in a CNT-based
oscillator, when the oscillator was subject to a finite
temperature. Such energy dissipation was found to be
strongly dependent on the morphology combination
of the tubes [32, 33], and the relative velocity of the
inner and outer tubes [34].
As employed in modeling and analysis of many
engineering designs, MD has difficulties in
simulating large models of CNT-based oscillators.
However, the existing continuum approaches and
multiscale methods cannot directly be employed. In
this paper, we propose a nanoscale continuum model
of CNT-based oscillators and study the mechanisms
of oscillators as well as their applications as memory
cells. Consequently, this research will contribute to
assist other engineering design of CNT-based
devices.
The outline of this paper is as below. Molecular
modeling and simulation of nanotube-based
oscillators are reviewed in Section 2. In Section 3, a
nanoscale continuum modeling is proposed, and it is
employed to study CNT-based oscillators in Section
4. Then, a NEMS design consisting of a nanotube-
based oscillator is described and studied as memory
cells in Section 5 followed by the conclusion.
II. MOLECULAR MODELING OF CNT-
BASED OSCILLATORS
A simple CNT-based oscillator is designed via a
double-walled carbon nanotube (DWNT), which
consists of an inner tube (core) and an outer tube
(shell), as shown in Fig. 1. When the outer tube is
partially fixed, the inner tube can oscillate inside the
outer tube once it is given an initial velocity or an
initial extrusion length.
Figure 1. A DWNT-based oscillator.
MD simulations of CNT-based oscillator were
conducted [31] via solving the following equations of
motion if there were no external forces:
 
I
LJouterinner
I
II
EEEE
m
dd
d





 (1)
where E is the total potential of the simulated
oscillator, which consists of the potential of the inner
tube, innerE , the potential of the outer tube, outerE ,
and the potential due to the interlayer interaction
LJE . Im is the mass of atom I and Id is its
displacement.
The reactive empirical bond order (REBO)
potential function [35] was used for describing
carbon-carbon interatomic interactions within a
single CNT. This potential function considers
chemical reactions and includes remote effects
caused by conjugated bonding, and it can be written
as follows:
    

i ij
ij
A
ijij
R
rVbrVE
)(
(2)
where ijr is the bond length,  rV R
and  rV A
are
pair-additive interactions that represent all
interatomic repulsions and attractions from valence
electrons, respectively. ijb contains the functions that
depend on the local coordinates and bond angles for
atoms i and j . The detailed formulations of REBO
can be found in [35]. The equilibrium carbon-carbon
bond length obtained by minimizing this potential is
nm142.0 .
The interlayer interaction between the inner tube
and the outer tube is described by the following
Lennard-Jones 6-12 potential [36],
 








 612
6
0 1
2
1
rr
y
ArELJ (3)

where 624
1043.2 JnmA 
 and nmy 3834.00  .
Minimizing the Lennard-Jones potential gives an
equilibrium interlayer distance of 0.34nm, which
results in stable oscillators as discussed by Legoas
and his co-workers [29].
As an example, we consider a (10,10)/(5,5)
double-walled nanotube as the oscillator, in which
the (5,5) inner tube has a length of 2.5 nm while the
(10,10) outer tube has a length of 3.7 nm. The initial
extrusion length is half of the inner tube length. Two
MD simulations are conducted. In the first study, the
oscillator is considered as an isolated system, i.e. no
heat exchanging between the oscillator and its
surrounding. The oscillation mechanisms of the
oscillator can be studied through the calculated
separation distance between the inner and the outer
tubes. It is found that the oscillation of the simulated
oscillator is stable as shown by the solid line in Fig.
2. The calculated oscillatory frequency is 55 GHz.
In the other simulation, the oscillator is at the
room temperature of 300K, and the Hoover
thermostat [37] is employed in the MD simulation.
As shown by the dashed line in Fig. 2, we find that a
stable oscillatory frequency and amplitude cannot be
obtained. The oscillatory amplitude decays until the
nano-oscillator eventually stops. This phenomenon is
caused by the interlayer friction, which mainly
dissipates the interlayer energy of the oscillator. Such
dissipated energy is transferred to the kinetic energy
of the outer tube, and the artificial thermostat
thereafter dissipates the energy of the outer tube. As
a result, the whole energy of the system is dissipated.
Figure 2. Oscillation mechanism of a (10,10)/(5,5)
DWNT-based oscillator.
III. NANOSCALE CONTINUUM MODELING
AND SIMULATION OF CNT-BASED
OSCILLATORS
III.1 CONTINUUM MODEL OF A SINGLE-WALLED
CARBON NANOTUBE
Since a single-walled carbon nanotube (SWNT)
can be viewed as an appropriate roll-up of a planar
graphene sheet, it is convenient to define the planar
graphene sheet as the undeformed or reference
configuration during the continuum modeling of a
SWNT. The continuum object replacing the
crystalline monolayer is a curvature surface without
thickness. Therefore, the nuclei are assumed to lie on
the surface, and the lattice vectors are chords of the
surface.
In most continuum models of nanomaterials, the
Cauchy-Born (CB) rule [38] was employed. The
Cauchy-Born hypothesis assumes that the lattice
vectors deform as line elements within a
homogeneous deformation, FAa  , where a and
A are deformed and undeformed lattice vectors
respectively. F is the deformation gradient which
maps the “infinitesimal” tangential material vectors
of the undeformed body into the vectors of the
deformed body. The lattice vectors a and A , each
of which connects two atomic positions, are physical
entities with finite length. Consequently, the CB rule
holds exactly, at least locally, only when the
“infinitesimal” material vector is equivalent to the
lattice vector. Therefore, the deformation is assumed
to be locally homogeneous.
Arroyo and Belytschko [39] introduced a so-
called “exponential Cauchy-Born rule” (ECB rule) to
overcome the difficulties of conventional CB rule in
approximating curved membranes. The exponential
mapping, which maps the lattice vector precisely
from the reference configuration to the current one, is
approximated locally via the ECB rule based on the
stretch and curvature of the membrane. In studying
mechanisms of CNT-based co-axial oscillators, we
are mainly interested in the longitudinal motion of
the CNT. We assume that the nanotube is deformed
uniformly in the radial direction so that the out-of-
plane motion can be neglected. Consequently, the CB
rule is still appropriate as a homogenization
technique in the continuum model of SWNTs. In this
case, the lattice vectors are approximated in the
tangent plane at the material point.
Since only the developable surface with zero
Gaussian curvature can be mapped to a planar body,
the caps of a CNT, which are semi spheres, cannot be
modeled via the proposed continuum approach.
However, the need to represent this kind of surfaces
can be covered by splitting the surface into pieces.
Those pieces can be represented as approximately
developable patches, and then glued together with
continuity boundary conditions, which are easily
accommodated by the Lagrange multiplier technique.
III.2 MESHFREE PARTICLE APPROXIMATION
Using the meshfree particle method [40], the
surface of a SWNT is discretized with a number of
particles. Since the out-of-plane motion of the
nanotube is ignored, the particles are only allowed to
move along the surface of the nanotube. It is natural
to use the Cartesian components on the planar
graphene in 2
 to express the particle coordinates in

3
 . If we denote x and gx as the Cartesian
components of particles in the nanotube and graphene
respectively, and R as the radius of the nanotube, the
mapping can be written as
 gg xxx   32
: (4)
or
,)/sin(
)/cos(
23
12
11
g
g
g
xx
RxRx
RxRx



(5)
In a planar graphene sheet, we define 2
gx
in the deformed/current configuration and 2
X in
the undeformed/reference configuration. According
to the meshfree particle methods [40], the
displacement field can be approximated by
     
I
II
h
tt uXXu , (6)
where  XI are called Lagrangian kernel functions.
The deformation gradient is then written as
XxF  g , in which h
g uXx  .
The nominal stress, P , can be computed as
  T
Cw
F
F
FP



)( (7)
where Cw is the strain energy density and calculated
as the potential density via the CB rule. It should be
noted that the free energy density will be employed in
equation 7 when considering the temperature effects
via the temperature-related Cauchy-Born rule [18].
The internal force in 2
 is then calculated as
 





0
0
int
dI
gI P
X
X
f

(8)
which can be mapped onto the nanotube as
 intint
gII ff  (9)
Indeed, the equations of motion solved in 3

are
int
I
ext
IIIm ffu  (10)
where Im and ext
If are the mass and the external
force of particle I, respectively.
III.3 NON-BONDED INTERLAYER INTERACTION
The internal forces calculated above are due to
the deformation of the nanotube itself. In the MD
simulation of CNT-based oscillators, the non-bonded
interlayer interaction, as in equation 3, plays an
important role. Although a SWNT or single layer of a
MWNT can be modeled by the continuum
approximation with the meshfree particle method as
described in the above, the continuum approach to
interlayer interaction, Lennard-Jones interaction,
remains an issue. In the meshfree particle method-
based continuum model, this issue becomes how to
map the Lennard-Jones interaction between two
groups of atoms to the interaction between two
particles.
To calculate the non-bonded energy in the
continuum level, two representative unit cells, one in
the inner tube and the other in the outer tube, of area
S0 are chosen. Each of the cells contains n nuclei
(n=2 for graphene). The continuum-level Lennard-
Jones energy density is
     dESnd LJLJ
2
0 (11)
where IOd xx  is the distance between the
centers of those two considered cells.
The total continuum-level non-bonded energy is
calculated as
   

O I
OIIOLJ ddxx (12)
where O and I are the deformed configuration
of outer and inner tube respectively. Then, the force
applied on particle I can be derived as the first
derivative of  with respect to the coordinates of
particle I . Such forces will be counted as the
external forces in solving the following equations of
motion,
int
I
I
IIm f
x
u 



(13)
III.3 NANOSCALE CONTINUUM SIMULATIONS
We restudy the oscillation mechanism of the (10,
10)/(5, 5) DWNT-based oscillator as shown in Fig.1
via the meshfree particle method. In the continuum
model shown in Fig. 3, the outer tube surface is
discretized with 320 particles, and the inner tube is
modeled by 152 particles.
Figure 3. The continuum model of a (10,10)/(5,5)
DWNT oscillator.
In the first simulation, the outer tube is fixed and
the inner tube is given an initial extrusion, which is
half of the inner tube length. When the inner tube is
released without any initial velocity, the interlayer
force, due to the Lennard-Jones energy between the
tubes, will drive the inner tube to move towards the
center of the outer tube. The inner tube will be
accelerated until the interlayer potential reaches the
minimum. The center-of-mass velocity of the inner
tube starts to decrease when the interlayer potential
increases. Once the center-of-mass velocity of the
inner tube becomes zero, the separation reaches the

maximum. Then the interlayer force tends to drive
the inner tube backwards to the center of the outer
tube. Fig. 4 shows the evolution of separation
distance obtained from the meshfree particle method
and compared well with the molecular dynamics
result. The calculated oscillatory frequency is around
55GHz which agrees to molecular dynamics
simulations [31] as well.
Figure 4. Separation distance evolutions.
Figure 5. Relative velocity contours on the inner tube
at various times.
In this simulation, we observe that shock waves
propagate along the inner tube when it oscillates
inside the outer tube. Fig. 5 illustrates the relative
velocities, which are the particle velocities with
respect to the local coordinate system at mass center
of the inner tube. It can be seen that several shock
waves are generated on the inner tube during its
oscillation. Those shock waves propagate back and
forth between the ends of the inner tube and interact
with each other. However, the relative velocities are
very small in comparison with the velocity of the
center-of-mass of the inner tube. It should be noted
that the background meshes of the outer tube are
plotted in Fig. 5 for visualization.
Figure 6. The effects of outer tube length (top) and
inner tube length (bottom) on the oscillator
frequency.
We also investigate the effect of tube length on
the frequency of CNT-based oscillators. We first
study (10,10)/(5,5) DWNT-based oscillators with
various length of the outer tube. The inner tube has a
constant length of 2.5 nm and the initial extrusion is
half of its length as described in the above. Fig.6
(top) shows that the oscillatory frequency is lower
with a longer outer tube. It agrees well with MD
simulations as well as the estimation in [31]. To
study the effect of the inner tube length on the
frequency of oscillators, we use the (10,10) outer
tube with the length of 25 nm. Fig.6 (bottom)
concludes that the frequency becomes lower with a
longer inner tube.
The interlayer friction was observed in the nano
oscillator at a finite temperature via MD simulations
[31]. In the nanoscale continuum simulation, the
interlayer friction is prescribed as the external force
when solving the equations of motion. It has been

shown in [31] that the interlayer friction depends on
the temperature as well as the chirality difference of
the inner/outer tubes. However, the calculated
interlay frictions have the same order as what
Cumming and Zettles [41] predicted, i.e. 0.015 pN
per atom. In this paper, we mainly study the
mechanisms of CNT-based oscillators at the room
temperature of 300K, and the interlayer friction of
0.025 pN per atom at the room temperature is
employed based on MD simulations. Therefore, the
equations of motion, i.e. equation 13, solved in
nanoscale continuum simulations are modified as
  int
9025.0 Iz
Iz
Iz
I
LJ
II
v
v
Nem fe
x
u 


 (14)
where N is the number of atoms that one particle
represents in the nanoscale continuum model.
We restudy the oscillation of the (10,10)/(5,5)
DWNT-based oscillator at 300K. The energy
dissipation is observed and the oscillation is getting
weaker and weaker till ceasing as shown in Fig. 7. It
compares well with the MD simulation results in
[31].
Figure 7. Separation distance evolutions when the
oscillator is at the room temperature of 300K.
IV. CNT-BASED MEMORY CELLS
IV.1 A CNT-BASED NEMS SYSTEM
Since the CNT-based oscillator's oscillatory
motion is not stable at a finite temperature, a NEMS
is designed so that stable oscillation can be achieved.
The conceptual idea is shown in Fig. 8. A DWNT is
positioned on top of a conducting ground plane. It
should be noted that the outer tube is semiconducting
while the inner tube can be either metallic or
semiconducting. For example, a (17,0)/(5,5) DWNT
can be used here since (17,0) CNTs are
semiconducting while (5,5) CNTs are metallic. The
atomic materials for conducting electrodes 1 and 2
are deposited on top of the outer nanotube. In this
configuration, the inner tube sits in a double-bottom
electromagnetic potential well. The depth of the
potential well under electrode 1 is proportional to the
voltage applied to electrode 1. Similarly, the depth of
the potential well under electrode 2 is proportional to
the voltage applied to electrode 2. The
electromagnetic forces exist when applying a high
voltage to one electrode while a low voltage is
applied to the other electrode. Such electromagnetic
forces can overcome the interlayer friction. The high
applied voltage is referred to as the WRITE voltage,
and the low applied voltage is the READ voltage.
Consequently, the inner tube will move from one side
to another due to the induced electromagnetic forces,
and the stable oscillation could be obtained. In most
cases, the outer tube is capped so that the inner tube
won’t escape if the induced electromagnetic forces
are too large.
Figure 8. A CNT-based NEMS.
It shall be noted that the capacitance of the
NEMS gate can be read by a distinct READ process.
A constant-current pulse, i.e. a READ voltage, is
applied to one of the electrodes. If the inner CNT is
present under that electrode, a relatively large
capacitance will be observed, and the time required to
charge the electrode will be longer. If the inner tube
is not present under that electrode, a relatively small
capacitance come forth with a concomitant fast
charging time for the electrode. As a result, the logic
state of the NEMS gate can be determined. It is
worthy to mention that all READ voltages are
sufficiently small so that the motion of the inner tube
will not be influenced. Whether the inner tube is
underneath electrode 1 or electrode 2 will result in
two different physical states determined by the
READ voltage. These two states can be interpreted as
Boolean logic states. Therefore, the system can be
used as a random access memory (RAM) cell.
IV.2 COMPUTATIONAL MODEL
The similar nanoscale continuum modeling can
be employed to study the proposed CNT-based
NEMS. However, the induced electromagnetic forces
need to be included as part of external forces when
solving the equations of motion.
In the proposed NEMS design, the outer CNT is
semiconducting so that its electric property is very
similar to that of an insulator. Consisting of two
electrodes and the conducting ground plane, the
whole device can be viewed as a two-conductor
capacitor with the dielectric spaces between them.
The capacitance of the system varies with the
presence or absence of the inner CNT. In other
words, the electric energy stored in the system varies
with different positions of the inner tube when a
constant voltage is applied on the electrode.
Therefore, the inner tube will be driven by the

induced electromechanical force due to the gradient
of the electric energy. Poisson’s equation is a general
way to calculate the electric potential for a given
charge distribution, and then the electric field can be
obtained
0
2
/  , Ε (15)
where  is the potential (in volts),  is the charge
density (in coulombs per cubic meter),
mFe /12854.80  is the permittivity of free
space (in farads per meter), and Ε is the electric
field. Then, the capacitance, C , can be expressed as
VdSC
S





   0Ε (16)
where S is for a surface integral over the conductor,
and V is the potential difference between the
electrode and the ground plane.
If the length of the electrode along the longitude
direction of the nanotube is sufficiently large as
regards with the diameter of the outer nanotube, it
can be assumed that the electric field is uniformly
distributed along the longitude direction and the
fringing regions at the both ends of the electrode are
negligible. Here, two cases are considered to
calculate the capacitance: the capacitor with or
without the presence of the inner tube, respectively.
For each case, the electric field and the capacitance
can be easily calculated by the two-dimensional
model. If the inner tube is partially under the
electrode, the system is effectively the combination
of two capacitors in parallel,
i.e.
 
tubeinnerwithout
tubeinnerofpart
tubeinnerwith
lC
llClC
lC
C
t
tt
t






00
0100
01
(17)
where t
C1 and t
C0 are the capacitances per unit
length with and without inner tube, respectively. l0 is
the inner tube length. l is the length of the inner tube
underneath the electrode, and it varies with the
positions of the inner tube, i.e.  zfl  . Hence, the
induced electromagnetic force applied on the inner
tube can be calculated as
 
z
C
VzF


 2
2
1
(18)
and the equations of motion, equation 14, at the room
temperature can be modified as
 
int2
2
1
9025.0
Iz
I
z
Iz
Iz
I
LJ
II
z
C
V
v
v
Nem
fe
e
x
u








(19)
Due to the Gauss’s law, the calculation of the
capacitance should be independent of the potential
and the total charge. As an example to calculate the
capacitance, a (17,0)/(5,5) DWNT with one electrode
of 10V is consider as shown in Fig. 9.
r = 0.34
R
=0.68
r = 0.32
l = 0.55
L = 3.0
Inner Tube
Outer Tube
Ground Plane
Electrode
Free Space
Unit: nm
Figure 9. Cross-sectional view of the considered
NEMS.
Figure 10. The contours of the electric fields when
the inner tube is absent (top) or present (bottom).
Fig. 10 shows the electric fields when the inner
tube is absent (top) or present (bottom). The
calculated capacitances are Fe 1903.1  and

Fe 1911.1  , respectively. It should be noted that if
the diameter of the outer nanotube is large and the
length of the electrode is short, the fringing regions
cannot be neglected. In that case, a three-dimensional
calculation would be appropriate.
IV.3 SIMULATIONS AND RESULTS
At first, a (17,0)/(5,5) DWNT-based NEMS as
the memory cell is considered. In this device, the
(17,0) outer CNT has a length of 6.4nm, and the (5,5)
inner has a length of 3.7 nm. The electrodes attached
to the outer tube are 2.0nm in length and 1.1nm in
width. Initially the inner tube is located at the center
of the outer tube. In the nanoscale continuum
simulations, this device is modeled via 884 particles.
It has been found that the voltage applied on the
electrodes cannot be too large; otherwise the CNT
could start unraveling carbon chains from the
exposed edge. Lee et al. [42] claimed that the
breakdown voltage of the CNT is round 2.0V/Å.
Therefore, the CNTs employed in this NEMS could
afford a potential of several hundred volts to maintain
its electrochemical stability. Here we apply 16V
constant voltage on the two electrodes alternatively
with a time interval of 1ns.
Figure 11: Boolean logic states of the (17,0)/(5,5)
CNT-based SRAM cell.
At the beginning, a WRITE voltage is applied on
electrode 2. The inner tube is stimulated to move due
to the induced electromagnetic force. Such an
electromagnetic force overcomes the interlayer
friction, and the inner tube moves to be underneath
electrode 2. After the inner tube is bounced back by
the cap of the outer tube, the interlay friction and the
induced electromagnetic force will resist its
movement towards to electrode 1. This phenomenon
results in the inner tube oscillating underneath
electrode 2 till ceasing. The position of the inner tube
can be detected by the READ process and the logic
state 1 is produced. When the WRITE voltage is
shifted to electrode 1, the inner tube moves again and
halts under electrode 1. The READ process can
detect its position and the logic state 0 is produced as
shown in Fig. 11. We also studied CNT-based
memory cells with long tubes and observed similar
phenomena. It can be seen that the frequency of the
memory cell depends on the frequency of the voltage
shifting, which means the device working as a
SRAM. It can be calculated that the frequency of this
SRAM is 500MHz according to the time interval of
the WRITE voltage shifting. Fig. 11 also shows that
it takes 500ps for the inner tube to cease under an
electrode, the recommended maximum frequency is
1GHz.
The proposed memory cell was designed on the
basis of the CNT-based oscillator, which has its own
natural frequency. The application of the proposed
design can be extended to a dynamic random access
memory (DRAM) cell if its embedded oscillator can
oscillate at its natural frequency. As being studied
before, a CNT-based oscillator is an underdamped
system, and its oscillation will eventually cease due
to the interlay friction when the oscillator is subject
to a finite temperature in practical usages. To ensure
the oscillator to oscillate consistently at its natural
frequency, a WRITE voltage pulse is applied every
several oscillation periods to stimulate the oscillator.
The period of the WRITE voltage pulse should not be
larger than natural period of the oscillator so that the
induced voltage acts to enhance oscillation when it
becomes weak. Consequently, a steady oscillation
can be generated.
Figure 12. Boolean logic states of the nanotube-based
DRAM memory cell.
A memory cell containing a (17,0)/(5,5) CNT-
based oscillator as the DRAM is studied here. The
(17,0) outer tube has a length of 32 nm while the
(5,5) inner tube has a length of 18 nm. The natural
oscillating frequency of the oscillator is 6.75 GHz
based on the nanoscale continuum simulation. The
open-ended outer tube is employed to ensure that the
inner tube can oscillate at the natural frequency of the

oscillator. Two 10-nm-long electrodes are attached
on the top of the outer tube. A WRITE voltage of 48
V with a duration of 2 ps is applied on electrode 2 so
that the inner tube starts to move. Then, the
amplitude of the oscillation will decrease due to the
interlayer friction. At every four oscillation cycles,
the same WRITE voltage pulse is given on electrode
2 to stimulate the oscillation. The WRITE voltage is
only applied on electrode 2 in this case to enhance
the oscillation from time to time. However, the
READ voltage is applied on both electrodes 1 and 2
to generate the logic states 0 and 1 as shown in Fig.
12.
V. CONCLUSION
A nanoscale continuum model was proposed to
study the CNT-based oscillators via the meshfree
particle method. Both the oscillators as isolate
systems and the ones at a finite temperature were
studied, and the results agreed well with the MD
simulation results. The shock wave propagation and
interaction were observed along the inner tube during
the oscillatory motion. Since the wave amplitude was
small, it didn’t have effects on oscillating
mechanisms of the oscillator. The proposed
numerical model could be referred to as a framework
to model and simulate other nanoscale devices.
In addition, a design of NEMS, containing a
CNT-based oscillator has been proposed as a memory
cell. The electrodes were attached on the top of the
outer tube so that the induced electromagnetic force
could overcome the interlayer friction when a
WRITE voltage was applied on the electrode. On the
other hand, a READ voltage was applied to detect the
position of the inner tube so that the Boolean logic
states could be generated. Numerical simulations
showed that the designed NEMS could be employed
as either SRAM or DRAM cells.
VI. Acknowledgements
Zhang acknowledges support from NSFC
(11272192).
REFERENCES
[1] S. Iijima, Helical microtubules of graphitic
carbon, Nature, 354, 1991 56-58.
[2] V. N. Popov, Carbon nanotubes: properties
and application , Materials Science and
Engineering R – Reports, 43(3), 2004, 61-
102.
[3] P. H. Li, W. J. Zhang, Q. F. Zhang, and J. L.
Wu, Nanoelectronic logic circuits with
carbon nanotube transistors, ACTA Physica
Sinica, 56(2), 2007, 1054-1060.
[4] Y. A. Tarakanov, and J. M. Kinaret, A
carbon nanotube field effect transistor with a
suspend nanotube gate, Nano Letters, 7(8),
2007, 2291-2294.
[5] Y. E. Lozovik, A. G. Nikolaev, and A. M.
Popov, Nanotube-based
nanoelectromechanical systems, Journal of
Experimental and Theoretical Physics,
103(3), 2006, 449-462.
[6] J. Del Nero, F. M. de Souza, R. B. Capaz,
Molecular electronics devices: a short
review, Journal of Computational and
Theoretical Nanoscience, 7(3), 2010, 503-
516.
[7] A. D. Franklin, R. A. Sayer, T. D. Sands, D.
B. Janes, and T. S. Fisher, Vertical carbon
nanotube devices with nanoscale lengths
controlled without lithography, IEEE
Transactions on Nanotechnology, 8(4),
2009, 469-476.
[8] D. Srivastava, A phenomenological model
of the rotation dynamics of carbon nanotube
gears with laser electric fields,
nanotechnology, 8, 1997, 186-192.
[9] J. W. Kang and H. J. Hwang, Nanoscale
carbon nanotube motor schematics and
simulations for micro-electro-mechanical
machines, Nanotechnology, 15, 2004, 1633-
1638.
[10] A. M. Popov, E. Bichoutskaia, Y. E.
Lozovik, and A. S. Kulish,
Nanoelectromechanical systems based on
multi-walled nanotubes: nanothermometer,
nanorelay, and nanoactuator, Physica Status
Solidi A – Applications and Materials
Science, 204(6), 2007, 1911-1917.
[11] W. S. Zhao, G. Agnus, V. Derycke, A.
Filoramo, J. P. Bourgoin, and C. Gamrat,
Nanotube devices based crossbar
artchitecture: toward neuromorphic
computing, Nanotechnology, 21(17), 2010,
175202.
[12] S. Axelsson, E. E. B. Campbell, L. M.
Jonsson, J. Kinaret, S. W. Lee, Y. W. Park,
and M. Sveningsson, Theoretical and
experimental investigations of three-
terminal carbon nanotube relays, New
Journal of Physics, 7, 2005, 245.
[13] C. H. Ke and H. D. Espinosa, Feedback
controlled nanocantilever device, Applied
Physics Letters, 85(4), 2004, 681-683.
[14] C. H. Ke, H. D. Espinosa, N. Pugno,
Numerical analysis of nanotube based
NEMS devices – Part II: Role of finite
kinetics, stretching and charge
concentrations, Journal of Applied
Mechanics – Transactions of the ASME,
72(5), 2005, 726-731.
[15] E. Bichoutskaia, Modelling interwall
interactions in carbon nanotubes:
fundamentals and device applications,

Philosophical Transactions of the Royal
Society, 365, 2007, 2893-2906.
[16] M. Poot, B. Witkamp, M. A. Otte, and H. S.
J. van der Zant, Modelling suspended carbon
nanotube resonators, Physica Status Solidi B
– Basic Solid State Physics, 244(11), 2007,
4252-4256.
[17] J. Marian, G. Venturini, B. L. Hansen, J.
Knap, M. Ortiz, and G. H. Campbell, Finite-
temperature extension of the
quasicontinuum method using Langevin
dynamics: entropy losses and analysis of
errors, Modelling and Simulation in
Materials Science and Engineering, 18(1),
2010, 015003.
[18] S. P. Xiao and W. X. Yang, A temperature-
related homogenization technique and its
implementation in the meshfree particle
method for nanoscale simulations,
International Journal for Numerical
methods in Engineering, 69(10), 2007,
2099-2125.
[19] S. Q. Chen, W. N. E and C. W. Shu, The
heterogeneous multiscale method based on
the discontinuous Galerkin method for
hyperbolic and parabolic problems,
Multiscale modeling and simulation, 3(4),
2005, 871-894.
[20] N. Choly, G. Lu, W. E and E. Kaxiras,
Multiscale simulations in simple metals: A
density-functional-based methodology,
Physical Review B, 71(9), 2005, 094101.
[21] G. J. Wagner and W. K. Liu, Coupling of
atomic and continuum simulations using a
bridging scale decomposition, Journal of
Computational Physics, 190, 2003, 249-274.
[22] S. P. Xiao and T. Belytschko, A bridging
domain method for coupling continua with
molecular dynamics, Computer Methods in
Applied Mechanics and Engineering, 193,
2004, 1645-1669.
[23] J. W. Kang, S. C. Kong, H. J. Hwang,
Electromechanical analysis of suspended
carbon nanotubes for memory applications,
Nanotechnology, 17, 2006, 2127-2134.
[24] M. Dequesnes, Z. Tang, and N. R. Aluru,
Static and dynamic analysis of carbon
nanotube-based switches, Transactions of
the ASME, 126, 2004, 230-237.
[25] S. P. Xiao and W. Y. Hou, Studies of
nanotube-based resonant oscillators via
multiscale modeling and simulation,
Physical Review B, 75, 2007, 125414.
[26] T. Belytschko and S. P. Xiao, Coupling
methods for continuum model with
molecular model, International Journal for
Multiscale Computational Engineering,
1(1), 2003, 115-126.
[27] Q. S. Zheng and Q. Jiang, Multiwalled
carbon nanotubes as gigahertz oscillators,
Physical Review Letters, 88(4), 2002,
045503.
[28] Q. S. Zheng, J. Z. Liu and Q. Jiang, Excess
van der Waals interaction energy of a
multiwalled carbon nanotube with an
extruded core and the induced core
oscillation, Physical Review B, 65, 2002,
245409.
[29] S. B. Legoas, V. R. Coluci, S. F. Braga, P.
Z. Coura, S. O. Dantas, and D. S. Galvao,
Molecular-dynamics simulations of carbon
nanotubes as gigahertz oscillators, Physical
Review Letters, 90(5), 2003, 055504.
[30] S. B. Legoas, V. R. Coluci, S. F. Braga, P.
Z. Coura, S. O. Dantas and D. S. Galvao,
Gigahertz nanomechanical oscillators based
on carbon nanotubes, Nanotechnology, 15,
2004, S184-S189.
[31] S. P. Xiao, D. R. Andersen, R. Han and W.
Y. Hou, Studies of carbon nanotube-based
oscillators using molecular dynamics,
International Journal of Computational and
Theoretical Nanoscience, 3, 2006, 142-147.
[32] W. L. Guo, Y. F. Guo, H. J. Gao, Q. S.
Zheng, and W. Y. Zhong, Energy dissipation
in gigahertz oscillators from multiwalled
carbon nonotubes, Physical Review Letters,
91(12), 2003, 125501.
[33] J. L. Rivera, C. McCabe and P. T.
Cummings, The oscillatory damped
behavior of incommensurate double-walled
carbon nanotubes, Nanotechnology, 16(2),
2005, 186-198.
[34] P. Tangney, S. G. Louie and M. L. Cohen,
Dynamic sliding friction between concentric
carbon nanotubes, Physical Review Letters,
93(6), 2004, 065503.
[35] D. W. Brenner, O. A. Shenderova, J. A.
Harrison, S. J. Stuart, B. Ni and S. B.
Sinnott, A second-generation reactive
empirical bond order (REBO) potential
energy expression for hydrocarbons, Journal
of Physics: Condensed Matter, 14, 2002,
783-802.
[36] L. A. Girifalco, R. A. Lad, Energy of
cohesion, compressibility and the potential
energy functions of the graphite system,
Journal of Chemical Physics, 25(4), 1956,
693-697.
[37] W. G. Hoover, Canonical dynamics -
Equilibrium phase-space distributions,
Physical Review A, 31, 1985, 1695-1697.
[38] J. Ericksen, The Cauchy and Born
hypotheses for crystals, In M. E. Curtin
(Ed.), Phase transformations and material
instabilities in solids, (Academic Press,

1984) 61-77.
[39] M. Arroyo and T. Belytschko, A finite
deformation membrane based on inter-
atomic potentials for the transverse
mechanics of nanotubes. Mechanics of
Materials, 35, 2003, 193-215.
[40] S. P. Xiao and W. X. Yang, A temperature-
related homogenization technique and its
implementation in the meshfree particle
method for nanoscale simulations,
International Journal for Numerical
Methods in Engineering, 69, 2007, 2099-
2125.
[41] J. Cumings and A. Zettl, Low-friction
nanoscale linear bearing realized from
multiwall carbon nanotubes, Science, 289,
2000, 602.
[42] Y. H. Lee, S. G. Kim and D. Tomanek,
Field-induced unraveling of carbon
nanotubes, Chemical Physics Letters, 265,
1997, 667-672.

Studying nanotube-based oscillators and their application as memory cells via nanoscale continuum modeling and simulation

Recommended

Recommended

More Related Content

Viewers also liked

Viewers also liked (20)

Similar to Studying nanotube-based oscillators and their application as memory cells via nanoscale continuum modeling and simulation

Similar to Studying nanotube-based oscillators and their application as memory cells via nanoscale continuum modeling and simulation (20)

Recently uploaded

Recently uploaded (20)

Studying nanotube-based oscillators and their application as memory cells via nanoscale continuum modeling and simulation